∫Calculus I Unit 1 – Functions and Graphs

Key Concepts

• Functions map input values from the domain to output values in the range
• The vertical line test determines if a relation is a function
• Piecewise functions consist of multiple sub-functions, each defined over a different interval of the domain
• Inverse functions "undo" each other, satisfying $f(f^{-1}(x))=f^{-1}(f(x))=x$
• Transformations of functions include shifts, reflections, and scaling
• Composition of functions combines two or more functions into a single function
• Limit notation describes the behavior of a function as the input approaches a certain value
• Continuity requires a function to be defined at a point, the limit to exist, and the limit to equal the function value

Function Basics

• A function $f$ is a rule that assigns a unique output $f(x)$ to each input $x$ in the domain
  • The domain is the set of all possible input values
  • The range is the set of all possible output values
• Functions can be represented using equations, graphs, or tables
• The vertical line test states that if any vertical line intersects a graph more than once, the graph does not represent a function
• One-to-one functions have a unique output for each input and pass the horizontal line test
• Even and odd functions exhibit symmetry about the y-axis and origin, respectively
  • Even functions satisfy $f(-x)=f(x)$
  • Odd functions satisfy $f(-x)=-f(x)$
• Increasing functions have outputs that increase as inputs increase, while decreasing functions have outputs that decrease as inputs increase

Types of Functions

• Linear functions have the form $f(x)=mx+b$, where $m$ is the slope and $b$ is the y-intercept
• Quadratic functions have the form $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants and $a \neq 0$
  • The graph of a quadratic function is a parabola
• Polynomial functions are the sum of terms with non-negative integer exponents (linear, quadratic, cubic)
• Rational functions are the quotient of two polynomial functions, often involving asymptotes and holes
• Exponential functions have the form $f(x)=a^x$, where $a>0$ and $a \neq 1$
• Logarithmic functions have the form $f(x)=\log_a(x)$, where $a>0$, $a \neq 1$, and $x>0$
  • Logarithmic functions are the inverses of exponential functions
• Trigonometric functions (sine, cosine, tangent) relate angles to side lengths in right triangles and have periodic behavior

Graphing Techniques

• To graph a function, plot points by evaluating the function at various input values
• Determine the domain and range of a function by considering the graph or equation
• Identify intercepts by setting $x=0$ (y-intercept) or $y=0$ (x-intercepts)
• Analyze asymptotes, which are lines that the graph approaches but never touches
  • Vertical asymptotes occur when the denominator of a rational function equals zero
  • Horizontal asymptotes describe the behavior of a function as $x$ approaches positive or negative infinity
• Use symmetry (even, odd) to simplify the graphing process
• Sketch piecewise functions by graphing each sub-function on its respective domain interval
• Utilize transformations to graph functions more efficiently

Transformations of Functions

• Transformations alter the graph of a function without changing its basic shape
• Vertical shifts move the graph up or down and have the form $f(x)+k$ (up for $k>0$, down for $k<0$)
• Horizontal shifts move the graph left or right and have the form $f(x-h)$ (right for $h>0$, left for $h<0$)
• Reflections flip the graph across the x-axis ($-f(x)$) or y-axis ($f(-x)$)
• Vertical scaling stretches ($|a|>1$) or compresses ($0<|a|<1$) the graph vertically and has the form $af(x)$
  • For $a<0$, the graph is also reflected across the x-axis
• Horizontal scaling stretches ($0<|a|<1$) or compresses ($|a|>1$) the graph horizontally and has the form $f(ax)$
• Combining transformations requires applying them in the correct order: scaling, reflection, horizontal shift, vertical shift

Function Operations

• Function addition $(f+g)(x)=f(x)+g(x)$ adds the output values of two functions for each input value
• Function subtraction $(f-g)(x)=f(x)-g(x)$ subtracts the output values of two functions for each input value
• Function multiplication $(fg)(x)=f(x)g(x)$ multiplies the output values of two functions for each input value
• Function division $(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$ divides the output values of two functions for each input value, provided $g(x) \neq 0$
• Composition of functions $(f \circ g)(x)=f(g(x))$ applies one function to the output of another function
  • The domain of the composite function is the set of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$
• Finding the inverse of a function $f^{-1}(x)$ involves solving the equation $y=f(x)$ for $x$ in terms of $y$, then swapping $x$ and $y$
  • The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$

Applications in Calculus

• Limits describe the behavior of a function as the input approaches a specific value or infinity
  • Limits are essential for defining continuity, derivatives, and integrals
• Continuity requires a function to have no gaps, jumps, or breaks in its graph
  • Continuous functions are necessary for many theorems and applications in calculus
• Derivatives represent the instantaneous rate of change of a function at a given point
  • Derivatives are used to find slopes of tangent lines, optimize functions, and analyze motion
• Integrals calculate the area under a curve, which can represent quantities such as displacement, work, and volume
  • The Fundamental Theorem of Calculus connects derivatives and integrals, enabling the calculation of areas and volumes

Common Pitfalls and Tips

• Remember that not all relations are functions; always check using the vertical line test
• Be careful when applying transformations; follow the correct order and sign conventions
• When finding the domain of a rational function, set the denominator equal to zero and exclude those values from the domain
• Ensure that the domain and range of a function align with its context in real-world applications
• When composing functions, work from the inside out, and pay attention to the order of the functions
• Remember that inverse functions "undo" each other; if $f(a)=b$, then $f^{-1}(b)=a$
• Sketch a graph to visualize a problem before attempting to solve it algebraically
• Double-check your work by plugging your answer back into the original equation or problem